Solution by Completing the Square

To solve $ax^2 + bx + c = 0$:

1. Divide by $a$: $x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0$

2. Move the constant: $x^2 + \dfrac{b}{a}x = -\dfrac{c}{a}$

3. Add $\left(\dfrac{b}{2a}\right)^2$ to both sides:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$

Example

Solve $x^2 + 4x - 5 = 0$ by completing the square.

$$x^2 + 4x = 5$$ $$x^2 + 4x + 4 = 5 + 4$$ $$(x + 2)^2 = 9$$ $$x + 2 = \pm 3$$ $$\therefore \; x = 1 \quad \text{or} \quad x = -5$$